How do you solve #1+ sqrt(x-3) = sqrt( x+2)#?

Question

Avery Alexander · Accepted Answer

#x=7# Square both sides #(1+sqrt(x-3))^2=x+2# #1+2sqrt(x-3)+cancel(x)-3=cancel(x)+2# #2=sqrt(x-3)# square both sides #x=7#

Eli Assouline · Answer

undefined To solve the equation 1 + sqrt(x-3) = sqrt(x+2), you can follow these steps:

1. Start by isolating one of the square roots on one side of the equation. In this case, subtract 1 from both sides to get sqrt(x-3) = sqrt(x+2) - 1.

2. Square both sides of the equation to eliminate the square roots. This gives you x - 3 = (sqrt(x+2) - 1)^2.

3. Expand the right side of the equation using the formula (a - b)^2 = a^2 - 2ab + b^2. In this case, a = sqrt(x+2) and b = 1. So, (sqrt(x+2) - 1)^2 = (sqrt(x+2))^2 - 2(sqrt(x+2))(1) + 1^2.

4. Simplify the equation further. The square of a square root cancels out, so you have x - 3 = (x+2) - 2(sqrt(x+2)) + 1.

5. Combine like terms and simplify the equation. This gives you x - 3 = x + 3 - 2(sqrt(x+2)).

6. Now, isolate the square root term by moving all other terms to one side of the equation. Subtract x from both sides to get -3 = 3 - 2(sqrt(x+2)).

7. Simplify further to obtain -3 = -2(sqrt(x+2)).

8. Divide both sides of the equation by -2 to solve for the square root term. This gives you sqrt(x+2) = 3/2.

9. Square both sides of the equation to eliminate the square root. This results in x + 2 = (3/2)^2.

10. Simplify the equation further. (3/2)^2 = 9/4, so x + 2 = 9/4.

11. Finally, solve for x by subtracting 2 from both sides of the equation. This gives you x = 9/4 - 2.

Therefore, the solution to the equation 1 + sqrt(x-3) = sqrt(x+2) is x = 1/4.